Direction of dipole moment and electric fields seem to create an issue

Question

1. If the direction of electric field is from positive to negative, how is it considered that when two equal (in terms of magnitude) but opposite charges are placed at a distance of $r$, the the dipole moment will be $ql$? My doubt lies in the fact that the force in the field is from postive to negative and from that, it seems that there will be no rotation or torque. But still, we see that there is a rotation about the centre of the two charges. Apparently, some force acts to turn them in different directions.

2. Secondly, why is the dipole moment from negative to the positive charge? (Please don't just say- "it's conventional. Kindly explain the reason behind that. Like, in the field, we can say that the field is from positive to negative because we consider a positive test charge and so, the direction it follows will be from positive to negative which comes from the fact that line charges repel and unlike ones attract each other. Kindly mention the story behind the convention because of which dipole moment is considered from negative to positive)

   I think I'm messing up with the direction of force somehow.

Answer 1

Your confusion relates to an insufficiently general definition of "moment." In the most general sense, a moment is a number that describes the shape of a distribution (of probability, mass, charge, force, etc.). In general, the $n$th-order statistical moment of a distribution is defined as

$$\langle r^n\rangle=\int r^n\rho(\vec{r})d^3\vec{r}$$

If $\rho$ is a probability distribution, the zeroth moment is 1, the first moment is the mean, the second moment is the variance, and so on. If $\rho$ is a mass distribution, the zeroth moment is the total mass, the first moment is the location of the center of mass multiplied by the total mass, and the second moment is the moment of inertia. For a force density distribution, the first moment of the distribution is called the torque (or simply the "moment" in engineering circles, which is short for "moment of force," though that usage is precisely what's confusing you).

In three dimensions, in order for the set of all moments to carry all of the information about the shape of a distribution, we must introduce some angular dependence to our expression. When we expand our potential into terms dependent on increasing powers of $r$ and the spherical harmonics (which carry angular information), this gives us the multipole moments $Q_{lm}$; reading them off of the expansion of the potential gives

$$Q_{lm}=\int r^l \rho(\vec{r})\sqrt{\frac{4\pi}{2l+1}}Y_{lm}^*(\theta,\phi)d^3\vec{r}$$

where $Y_{lm}$ are the spherical harmonics. The moments of order $l$ give the $2^l$-pole moment, projected onto various axes, where $-l\leq m \leq l$. For $l=0$, you get the monopole moment, which only has $m=0$ and is therefore just a number. For $l=1$, you get the dipole moment. Since $m=-1,0,1$, there are three components to the dipole moment, which means it exists as a vector. The direction of this vector is fixed by the definition of the multipole moment, and if you go through the necessary calculations, you will find that this definition fixes the dipole moment vector to be from the negative charge to the positive charge.

You can keep going up in $l$, too. For $l=2$, the quadrupole moment has 5 projections, for $m=-2,-1,0,1,2$. As such, the quadrupole moment turns out to be a traceless symmetric rank-2 tensor (essentially a matrix). Higher moments have even more complicated structures.

Answer 2

The strength of the electric field around a dipole depends on the product of the charge $q$ and its separation $d$, called the dipole moment $p=qd$. Even though the charges and separations are different, two dipoles which have the same product $qd$ - ie the same dipole moment - have the same electric field around them.

(This is approximately true provided that the distance from the dipole at which the electric field is measured is large compared with the separation $d$.)

Dipole moment is not a torque, although a dipole does look similar to a couple, and the use of the term moment is the same as for moment of a force and moment of inertia. If placed in a uniform electric field, there would be a torque on the dipole.

The direction of the dipole moment is the direction of the electric field coming out from the +ve end. This follows the convention that the electric field is the direction of the force on a small +ve test charge. The test charge cannot be placed inside the dipole, because this is usually very small, and in this region the above approximation does not apply so the definition of the dipole moment would not apply.